Large traveling capillary-gravity waves for Darcy flow

We study capillary-gravity and capillary surface waves for fluid flows governed by Darcy's law. This includes flows in vertical Hele-Shaw cells and in porous media (the one-phase Muskat problem) with finite or infinite depth. The free boundary is acted upon by an external pressure posited to be in traveling wave form with an arbitrary periodic profile and an amplitude parameter. For any given wave speed, we first prove that there exists a unique local curve of small periodic traveling waves corresponding to small values of the parameter. Then we prove that as the parameter increases but could possibly be bounded, the curve belongs to a connected set $\mathcal{C}$ of traveling waves. The set $\mathcal{C}$ contains traveling waves that either have arbitrarily large gradients or are arbitrarily close to the rigid bottom in the finite depth case. To the best of our knowledge, this is the first construction of large traveling surface waves for a viscous free boundary problem.Comment: 25 page

Traveling wave solutions to the one-phase Muskat problem: existence and stability

We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.Comment: Funding source for H. Q. N. correcte

Slowly traveling gravity waves for Darcy flow: existence and stability of large waves

We study surface gravity waves for viscous fluid flows governed by Darcy's law. The free boundary is acted upon by an external pressure posited to be in traveling wave form with a periodic profile. It has been proven that for any given speed, small external pressures generate small periodic traveling waves that are asymptotically stable. In this work, we construct a class of slowly traveling waves that are of arbitrary size and asymptotically stable. Our results are valid in all dimensions and for both the finite and infinite depth cases.Comment: 22 page